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This definition tacitly includes a finiteness condition. A basis of a vector space is sometimes called a Hamel basis in order to distinguish it from the concept of an orthonormal basis of a Hilbert space and some other kinds of bases that occur in Banach spaces. An orthonormal basis of a Hilbert space H is an orthonormal set of members of H such that any member of the H can be written as a linear combination of a possibly infinite set of members of the orthonormal basis. In order to speak of infinite linear combinations, one must have a notion of convergence. Since Hilbert spaces are metric spaces, such a notion is available and natural.
Using Zorn's lemma, one can show that:
Example I: Show that the vectors (1,1) and (1,2) form a basis for R^{2}.
Proof: We have to prove that these 2 vectors are both linearly independent and that they generate R^{2}.
Part I: To prove that they are linearly independent, suppose that there are numbers a,b such that:
Part II: To prove that these two vectors generate R^{2}, we have to let (a,b) be an arbitrary element of R^{2}, and show that there exist numbers x,y such that:
Example II: We have already shown that E_{1}, E_{2}, ..., E_{n} are linearly independent and generate R^{n}. Therefore, they form a basis for R^{n}.
Example III: Let W be the vector space generated by e^{t}, e^{2t}. We have already shown they are linearly independent. Then they form a basis for W.
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